Legendre singularities of sub-Riemannian geodesics

TL;DR

This paper provides a comprehensive local classification of Legendre singularities of sub-Riemannian geodesics on surfaces, revealing dualities and connections to pendulum motion.

Contribution

It offers the first complete local classification of Legendre singularities for sub-Riemannian geodesics on surfaces, including duality relations.

Findings

Complete classification of Legendre singularities for sub-Riemannian geodesics.

Identification of duality in Legendre singularities.

Connection to pendulum motion phenomena.

Abstract

Let $M$ be a surface with a Riemannian metric and $UM$ the unit tangent bundle over $M$ with the canonical contact sub-Riemannian structure $D$ on $UM$. In this paper, the complete local classification of singularities, under the Legendre fibration $UM$ over $M$, is given for sub-Riemannian geodesics of $(UM, D)$. Legendre singularities of sub-Riemannian geodesics are classified completely also for another Legendre fibration from $UM$ to the space of Riemannian geodesics on $M$. The duality on Legendre singularities is observed related to the pendulum motion.